130 research outputs found
Driving Markov chain Monte Carlo with a dependent random stream
Markov chain Monte Carlo is a widely-used technique for generating a
dependent sequence of samples from complex distributions. Conventionally, these
methods require a source of independent random variates. Most implementations
use pseudo-random numbers instead because generating true independent variates
with a physical system is not straightforward. In this paper we show how to
modify some commonly used Markov chains to use a dependent stream of random
numbers in place of independent uniform variates. The resulting Markov chains
have the correct invariant distribution without requiring detailed knowledge of
the stream's dependencies or even its marginal distribution. As a side-effect,
sometimes far fewer random numbers are required to obtain accurate results.Comment: 16 pages, 4 figure
Adaptive independent sticky MCMC algorithms
Monte Carlo methods have become essential tools to solve complex Bayesian inference problems in different fields, such as computational statistics, machine learning, and statistical signal processing. In this work, we introduce a novel class of adaptive Monte Carlo methods, called adaptive independent sticky Markov Chain Monte Carlo (MCMC) algorithms, to sample efficiently from any bounded target probability density function (pdf). The new class of algorithms employs adaptive non-parametric proposal densities, which become closer and closer to the target as the number of iterations increases. The proposal pdf is built using interpolation procedures based on a set of support points which is constructed iteratively from previously drawn samples. The algorithm’s efficiency is ensured by a test that supervises the evolution of the set of support points. This extra stage controls the computational cost and the convergence of the proposal density to the target. Each part of the novel family of algorithms is discussed and several examples of specific methods are provided. Although the novel algorithms are presented for univariate target densities, we show how they can be easily extended to the multivariate context by embedding them within a Gibbs-type sampler or the hit and run algorithm. The ergodicity is ensured and discussed. An overview of the related works in the literature is also provided, emphasizing that several well-known existing methods (like the adaptive rejection Metropolis sampling (ARMS) scheme) are encompassed by the new class of algorithms proposed here. Eight numerical examples (including the inference of the hyper-parameters of Gaussian processes, widely used in machine learning for signal processing applications) illustrate the efficiency of sticky schemes, both as stand-alone methods to sample from complicated one-dimensional pdfs and within Gibbs samplers in order to draw from multi-dimensional target distributions
Independent doubly Adaptive Rejection Metropolis Sampling
Adaptive Rejection Metropolis Sampling (ARMS) is a wellknown
MCMC scheme for generating samples from onedimensional
target distributions. ARMS is widely used within Gibbs sampling, where automatic and fast samplers are often
needed to draw from univariate full-conditional densities.
In this work, we propose an alternative adaptive algorithm
(IA2RMS) that overcomes the main drawback of ARMS (an uncomplete adaptation of the proposal in some cases), speeding
up the convergence of the chain to the target. Numerical
results show that IA2RMS outperforms the standard ARMS,
providing a correlation among samples close to zero
Speeding Up MCMC by Delayed Acceptance and Data Subsampling
The complexity of the Metropolis-Hastings (MH) algorithm arises from the
requirement of a likelihood evaluation for the full data set in each iteration.
Payne and Mallick (2015) propose to speed up the algorithm by a delayed
acceptance approach where the acceptance decision proceeds in two stages. In
the first stage, an estimate of the likelihood based on a random subsample
determines if it is likely that the draw will be accepted and, if so, the
second stage uses the full data likelihood to decide upon final acceptance.
Evaluating the full data likelihood is thus avoided for draws that are unlikely
to be accepted. We propose a more precise likelihood estimator which
incorporates auxiliary information about the full data likelihood while only
operating on a sparse set of the data. We prove that the resulting delayed
acceptance MH is more efficient compared to that of Payne and Mallick (2015).
The caveat of this approach is that the full data set needs to be evaluated in
the second stage. We therefore propose to substitute this evaluation by an
estimate and construct a state-dependent approximation thereof to use in the
first stage. This results in an algorithm that (i) can use a smaller subsample
m by leveraging on recent advances in Pseudo-Marginal MH (PMMH) and (ii) is
provably within of the true posterior.Comment: Accepted for publication in Journal of Computational and Graphical
Statistic
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